Let \(\mathcal {Q}=\frac{1}{m}\sum _{i=1}^m \delta _{x^{\prime }_i}\)  be a measure created by data \(x^{\prime }_1, . . . , x^{\prime }_m\) . Then, \(\inf _{Z:|Z| \le k}{\frac{1}{m} \sum _{i=1}^m d^2(x^{\prime }_i,Z)}\)  is equivalent to explore a discrete measure \(\mathcal {H}\)  including finite number of
support points and minimizing the following objective: \(\inf _{\mathcal {H} \in \mathcal {O}_k(\theta )}{ \sum _{i=1}^m W_2^2(\mathcal {H},\mathcal {Q})}.\)  This problem can also be thought of as a Wasserstein barycenter problem when \(l = 1\) . From this prospective, as denoted by [1]}, the algorithm for finding the Wasserstein barycenters is an alternative for the popular Loyd’s algorithm to find local minimum of the K-means objective. Thus, we adopt the algorithm introduced in [1]} used for computing the Wasserstein barycenters of empirical probability measures to explore the clusters underlying the unlabeled data (Appendix B).
